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Abstract

We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in three-valued logic. Any semantics in this family satisfies two important properties: (i) it extends the partial stable semantics for normal logic programs and (ii) total stable models are always minimal. We also give a specific instance of the semantics and show that it has several attractive features.

Citations

...d by transfinite iteration of S starting from the bottom element in the # p order which is (#, #). The standard partial approximating operator of the TP operator is Fitting's three-valued #P operator =-=[6]-=-. The Kripke-Kleene fixpoint of #P is equal to the Kripke-Kleene semantics of P [6]. Although partial stable models [14] are defined in a very di#erent way they do coincide with partial stable fixpoin...

...s giving precise relationship between our semantics and most of the previous proposals for stable model semantics for aggregate programs. This includes the stable semantics of weight constraint rules =-=[16]-=- used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog [8] and the dlv system [2], and our previous work on the ultimate semantics of aggregate progr...

...us proposals for stable model semantics for aggregate programs. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey =-=[11]-=- which is also used by A-Prolog [8] and the dlv system [2], and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining th...

... several attractive features. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone =-=[13, 15]-=- and stratified [1, 13] ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Ex...

... several attractive features. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone =-=[13, 15]-=- and stratified [1, 13] ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Ex...

...ics for aggregate programs. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog =-=[8]-=- and the dlv system [2], and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining the syntax and semantics of aggregate...

...ures. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone [13, 15] and stratified =-=[1, 13]-=- ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Example 1). The developme...

...tial stable models. Thus, our definition also covers these two semantics. The foundation of this work is the algebraic theory of approximating operators developed by Denecker, Marek, and Truszczynski =-=[3, 4]-=-. The theory studies approximations of the fixpoints of non-monotone lattice operators O : L # L. With any such operator O, it associates a family of approximating operators A : L 2 # L 2 on the produ...

...computation of the well-founded fixpoint based on the partial stable operator of #P is very similar to the bottom-up evaluation technique based on the doubled program by Kemp, Srivastava, and Stuckey =-=[10]-=-. 3.2 Partial Stable Models of Aggregate Programs We are now ready to define our semantics. Our goal is to define a partial approximating operator # aggr P of T aggr P which, for programs without aggr...

...ow aggregates in the heads of rules. We think that this is not a serious limitation because such programs can be translated to programs without aggregates in the heads by introducing additional atoms =-=[12]-=-. For simplicity of the presentation, we defined the semantics for a propositional language and set expressions of finite size. It can be easily extended to programs with variables by considering a su...

...tial stable models. Thus, our definition also covers these two semantics. The foundation of this work is the algebraic theory of approximating operators developed by Denecker, Marek, and Truszczynski =-=[3, 4]-=-. The theory studies approximations of the fixpoints of non-monotone lattice operators O : L # L. With any such operator O, it associates a family of approximating operators A : L 2 # L 2 on the produ...

...ams. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog [8] and the dlv system =-=[2]-=-, and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining the syntax and semantics of aggregate programs (Section 2). ...